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SYMMETRIC TENSOR DECOMPOSITION
"... We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented ..."
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We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank. 1.
Graph Expansion Analysis for Communication Costs of Fast Rectangular Matrix Multiplication
, 2012
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Efficient Algorithms for Path Problems in Weighted Graphs
, 2008
"... Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shorte ..."
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Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the allpairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions
Master’s Examination Committee:
"... Using KML files as encoding standard to explore locations, access and ..."
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Using KML files as encoding standard to explore locations, access and
Author manuscript, published in "Linear Algebra and Applications 433, 1112 (2010) 18511872" SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variab ..."
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Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasiHankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in nongeneric cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for fl ..."
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Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
An overview of the recent progress on matrix multiplication
, 2012
"... The exponent ω of matrix multiplication is the infimum over all real numbers c such that for all ε> 0 there is an algorithm that multiplies n × n matrices using at most O(nc+ε) arithmetic operations over an arbitrary field. A trivial lower bound on ω is 2, and the best known upper bound until recent ..."
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The exponent ω of matrix multiplication is the infimum over all real numbers c such that for all ε> 0 there is an algorithm that multiplies n × n matrices using at most O(nc+ε) arithmetic operations over an arbitrary field. A trivial lower bound on ω is 2, and the best known upper bound until recently was ω < 2.376 achieved by Coppersmith and Winograd in 1987. There were two independent improvements on ω, one by Stothers in 2010 who showed that ω < 2.374, and one by myself that ultimately resulted in ω < 2.373. Here I discuss the road to these improvements and conclude with some open questions. 1